Inspired by these findings, sophisticated optimization algorithms were
proposed for hybrid beamforming [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. In
[1]}, the authors leveraged the channel's sparsity and
proposed an orthogonal matching pursuit (OMP) aided algorithm. Based
on the OMP concept, the authors of  [2]}, [3]}, [4]}
developed a hybrid transceiver architecture relying on the minimum
mean square error (MMSE) criterion. In [5]}, the
authors formulated a matrix factorization problem and developed a
manifold optimization (MO) based hybrid beamforming algorithm, where
the unit-modulus constraint was considered as a Riemannian manifold.
In [6]}, a codebook-based hybrid beamforming design was
conceived for maximizing the system's spectral efficiency. As a
further development, the authors of [7]} conceived a
two-stage optimization algorithm based on the general
eigen-decomposition method and their work evolved further
in [8]} to the broadband scenario with the aid of MO.
